Aus Online Mathematik Brückenkurs 1

Looking first at \displaystyle \sqrt{18} this square root expression can be simplified by writing 18 as a product of its smallest possible integer factors

\displaystyle 18 = 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}

and then we can take the quadratic out of the square root sign by using the rule \displaystyle \sqrt{a^{2}b}=a\sqrt{b} (valid for non-negative a and b),

\displaystyle \sqrt{18} = \sqrt{2\cdot 3^{2}} = 3\sqrt{2}\,\textrm{.}

In the same way, we write \displaystyle 8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3} and get

\displaystyle \sqrt{8} = \sqrt{2\cdot 2^{2}} = 2\sqrt{2}\,\textrm{.}

All together, we get

\displaystyle \begin{align} \sqrt{18}\sqrt{8} &= 3\sqrt{2}\cdot 2\sqrt{2}\\[5pt] &= 3\cdot 2\cdot \bigl(\sqrt{2}\bigr)^{2}\\[5pt] &= 3\cdot 2\cdot 2\\[5pt] &= 12\,\textrm{.} \end{align}